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A253190 Triangle T(n, m)=Sum_{k=1..(n-m)/2} C(m+k-1, k)*T((n-m)/2, k), T(n,n)=1.

%I A253190 #16 Dec 13 2023 08:53:22
%S A253190 1,0,1,1,0,1,0,2,0,1,1,0,3,0,1,0,3,0,4,0,1,2,0,6,0,5,0,1,0,6,0,10,0,6,
%T A253190 0,1,3,0,13,0,15,0,7,0,1,0,11,0,24,0,21,0,8,0,1,5,0,27,0,40,0,28,0,9,
%U A253190 0,1,0,20,0,55,0,62,0,36,0,10,0,1
%N A253190 Triangle T(n, m)=Sum_{k=1..(n-m)/2} C(m+k-1, k)*T((n-m)/2, k), T(n,n)=1.
%F A253190 G.f.: A(x)^m=Sum_{n>=m} T(n,m)x^n, A(x)=Sum_{n>0} a(n)*x^(2*n-1), a(n) - is A000621.
%e A253190 1;
%e A253190 0, 1;
%e A253190 1, 0, 1;
%e A253190 0, 2, 0, 1;
%e A253190 1, 0, 3, 0, 1;
%e A253190 0, 3, 0, 4, 0, 1;
%e A253190 2, 0, 6, 0, 5, 0, 1;
%p A253190 A253190 := proc(n,m)
%p A253190     option remember;
%p A253190     if n = m then
%p A253190         1;
%p A253190     elif type(n-m,'odd') then
%p A253190         0 ;
%p A253190     else
%p A253190         add(binomial(m+k-1,k)*procname((n-m)/2,k),k=1..(n-m)/2) ;
%p A253190     end if;
%p A253190 end proc: # _R. J. Mathar_, Dec 16 2015
%o A253190 (Maxima)
%o A253190 T(n, m):=if n=m then 1 else sum(binomial(m+k-1, k)*T((n-m)/2, k), k, 1, (n-m)/2);
%Y A253190 Cf. A000621 (row sums), A003600, A253184, A253189.
%K A253190 nonn,tabl
%O A253190 1,8
%A A253190 _Vladimir Kruchinin_, Mar 24 2015